L(s) = 1 | + (−1.35 + 5.06i)2-s + (−10.5 − 6.10i)3-s + (−9.99 − 5.77i)4-s + (0.491 + 1.83i)5-s + (45.3 − 45.3i)6-s + (44.1 − 76.4i)7-s + (−16.5 + 16.5i)8-s + (34.0 + 59.0i)9-s − 9.97·10-s − 127. i·11-s + (70.4 + 122. i)12-s + (7.06 + 26.3i)13-s + (327. + 327. i)14-s + (6.00 − 22.4i)15-s + (−153. − 266. i)16-s + (−283. − 75.9i)17-s + ⋯ |
L(s) = 1 | + (−0.339 + 1.26i)2-s + (−1.17 − 0.678i)3-s + (−0.624 − 0.360i)4-s + (0.0196 + 0.0734i)5-s + (1.25 − 1.25i)6-s + (0.900 − 1.55i)7-s + (−0.258 + 0.258i)8-s + (0.420 + 0.729i)9-s − 0.0997·10-s − 1.05i·11-s + (0.489 + 0.847i)12-s + (0.0417 + 0.155i)13-s + (1.67 + 1.67i)14-s + (0.0266 − 0.0996i)15-s + (−0.600 − 1.04i)16-s + (−0.981 − 0.262i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 37 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.668 + 0.743i)\, \overline{\Lambda}(5-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 37 ^{s/2} \, \Gamma_{\C}(s+2) \, L(s)\cr =\mathstrut & (0.668 + 0.743i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{5}{2})\) |
\(\approx\) |
\(0.586826 - 0.261507i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.586826 - 0.261507i\) |
\(L(3)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 37 | \( 1 + (1.28e3 + 465. i)T \) |
good | 2 | \( 1 + (1.35 - 5.06i)T + (-13.8 - 8i)T^{2} \) |
| 3 | \( 1 + (10.5 + 6.10i)T + (40.5 + 70.1i)T^{2} \) |
| 5 | \( 1 + (-0.491 - 1.83i)T + (-541. + 312.5i)T^{2} \) |
| 7 | \( 1 + (-44.1 + 76.4i)T + (-1.20e3 - 2.07e3i)T^{2} \) |
| 11 | \( 1 + 127. iT - 1.46e4T^{2} \) |
| 13 | \( 1 + (-7.06 - 26.3i)T + (-2.47e4 + 1.42e4i)T^{2} \) |
| 17 | \( 1 + (283. + 75.9i)T + (7.23e4 + 4.17e4i)T^{2} \) |
| 19 | \( 1 + (131. + 491. i)T + (-1.12e5 + 6.51e4i)T^{2} \) |
| 23 | \( 1 + (262. - 262. i)T - 2.79e5iT^{2} \) |
| 29 | \( 1 + (-37.7 - 37.7i)T + 7.07e5iT^{2} \) |
| 31 | \( 1 + (-674. - 674. i)T + 9.23e5iT^{2} \) |
| 41 | \( 1 + (-980. - 566. i)T + (1.41e6 + 2.44e6i)T^{2} \) |
| 43 | \( 1 + (223. - 223. i)T - 3.41e6iT^{2} \) |
| 47 | \( 1 - 262.T + 4.87e6T^{2} \) |
| 53 | \( 1 + (1.78e3 + 3.08e3i)T + (-3.94e6 + 6.83e6i)T^{2} \) |
| 59 | \( 1 + (-6.07e3 - 1.62e3i)T + (1.04e7 + 6.05e6i)T^{2} \) |
| 61 | \( 1 + (1.14e3 - 307. i)T + (1.19e7 - 6.92e6i)T^{2} \) |
| 67 | \( 1 + (4.73e3 + 2.73e3i)T + (1.00e7 + 1.74e7i)T^{2} \) |
| 71 | \( 1 + (-2.81e3 + 4.88e3i)T + (-1.27e7 - 2.20e7i)T^{2} \) |
| 73 | \( 1 - 1.02e4iT - 2.83e7T^{2} \) |
| 79 | \( 1 + (1.06e3 + 3.99e3i)T + (-3.37e7 + 1.94e7i)T^{2} \) |
| 83 | \( 1 + (-978. - 1.69e3i)T + (-2.37e7 + 4.11e7i)T^{2} \) |
| 89 | \( 1 + (2.70e3 - 1.00e4i)T + (-5.43e7 - 3.13e7i)T^{2} \) |
| 97 | \( 1 + (-9.90e3 + 9.90e3i)T - 8.85e7iT^{2} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−15.90342899525153318378352560618, −14.32384249796897256480031530668, −13.38845591425387207697766173883, −11.54319771728525792331499491565, −10.82641392011606495199839026776, −8.552092967246485943571681209296, −7.20719920566491055655136429216, −6.47881418888510273155502807282, −4.92281791713064704753847940693, −0.56652524004842072433651007570,
2.05515698743712618463529180332, 4.57136207886810906383076927328, 6.00147328904504179473357652496, 8.654348588446316951790912394228, 9.984134964127690549108378216605, 10.97062448926076223929342834993, 11.92019655467156842148260656917, 12.52945118835655453997488855279, 14.91521398261964948439681718884, 15.75190681269546162190217763349